“…For some applications in computer algebra, number theory, mathematical physics, and experimental mathematics, it is necessary to compute integrals to an accuracy of hundreds of digits, and occasionally even tens of thousands of digits [2,3,9,23]. The Gauss-Legendre formula (3) achieves an accuracy of p bits using n = O(p) evaluation points if f is analytic on a neighborhood of (−1, 1), and if the neighborhood is large (that is, if the path of integration is well isolated from any singularities of f ), then the constants hidden in the O notation are close to the best achievable by any quadrature rule [22]. This quality is related to the fact that (3) maximizes the order of accuracy among n-point quadrature rules for integrating polynomials, being exact when f is any polynomial of degree up to 2n − 1; as a result, the accuracy is also excellent for analytic integrands that are well approximated by polynomials.…”